#6652382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,38)(29,42)(30,33)(32,41)(34,44)(35,51)(36,46)(37,54)(39,56)(40,50)(43,49)(45,52)(47,53)(48,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,31)(28,39)(29,37)(30,36)(32,48)(33,46)(34,47)(35,40)(38,56)(41,55)(42,54)(43,52)(44,53)(45,49)(50,51);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28,30,40)(26,32,33,49)(27,35,36,39)(29,43,44,41)(31,45,46,48)(34,50,42,38)(37,55,53,52)(47,56,54,51);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(27,36)(31,46)(35,39)(37,53)(45,48)(47,54)(51,56)(52,55);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(26,33)(27,36)(28,40)(34,42)(37,53)(41,43)(45,48)(51,56)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=h,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6559578);